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database-normalization3nfbcnf

Is this relation in 3NF as well as in BCNF?

Suppose a relation schema R(A,B,C) and the FDs are

{A -> B, B -> C}

So the superkeys are {A}, {A,B}

Now if we decompose it into 3NF it will be

R1(A,B) with FD {A -> B} and R2(B,C) with FD {B -> C}

Is it in BCNF? I can't determine. Since B was not a superkey in R does {B -> C} in R2 violates BCNF?

Solution

  • {AB} is a superkey, but it's not a candidate key. (It's not a minimal superkey.) The decomposition

    • R1(A B)
    • R2(B C)

    is in at least BCNF.

    Informally, a relation is in BCNF if every arrow is an arrow out of a candidate key. B is a candidate key in R2.

    The relation R is not in BCNF. The only candidate key in R is A; the FD B->C has an arrow that's not out of a candidate key.

    In truth, both R1 and R2 are much stronger than BCNF. They're both in 6NF.